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Towards classification of multiple-end solutions to the Allen-Cahn equation in $$\mathbb{R}^2$$. (English) Zbl 1262.35011
Summary: An entire solution of the Allen-Cahn equation $$\Delta u = f(u)$$, where $$f$$ is an odd function and has exactly three zeros at $$\pm 1$$ and $$0$$, e.g. $$f(u) = u(u^2 - 1)$$, is called a $$2k$$-ended solution if its nodal set is asymptotic to $$2k$$ half lines, and if along each of these half lines the function $$u$$ looks (up to a multiplication by $$-1$$) like the one dimensional, odd, heteroclinic solution $$H$$, of $$H'' = f(H)$$. In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions $$U$$ whose nodal lines are precisely the straight lines $$y = \pm x$$. We describe the connected components of the moduli space of $$4$$-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all $$4$$-ended solutions are continuous deformations of the saddle solution.

##### MSC:
 35B08 Entire solutions to PDEs 35Q79 PDEs in connection with classical thermodynamics and heat transfer 35J61 Semilinear elliptic equations